利用辅助数据提升低秩矩阵补全精度，可自动筛选有用来源数据。
Transfer Learning for Matrix Completion
- 基于源数据相关性设计迁移学习流程，结合先验知识选择优质源。
- 理论证明方法收敛速度更快，且达到最优下界，消除对数因子影响。
- 当源数据相关性未知时，能自动识别有效来源，适合高维矩阵补全场景。
本文研究矩阵补全场景下的知识迁移问题，旨在利用可用的辅助数据提升目标低秩矩阵的估计性能。我们提出一种在已知源数据优劣情况下的迁移学习方法，并分析其收敛速率，证明了其最小最大最优性。分析表明，当源矩阵与目标矩阵足够接近时，该方法优于仅使用目标数据的传统方法。特别地，我们引入 \\( \cite{brailovskaya2024universality} \\) 中的先进尖锐浓度不等式，消除了收敛速率中的对数因子，这对证明最小最大最优性至关重要。当源数据的相关性未知时，我们设计了一种高效的检测方法以识别信息量丰富的源数据，并建立了其选择一致性。通过模拟实验和真实数据分析验证了方法的有效性。
原文摘要 · Abstract (English)
In this paper, we explore the knowledge transfer under the setting of matrix completion, which aims to enhance the estimation of a low-rank target matrix with auxiliary data available. We propose a transfer learning procedure given prior information on which source datasets are favorable. We study its convergence rates and prove its minimax optimality. Our analysis reveals that with the source matrices close enough to the target matrix, out method outperforms the traditional method using the single target data. In particular, we leverage the advanced sharp concentration inequalities introduced in \cite{brailovskaya2024universality} to eliminate a logarithmic factor in the convergence rate, which is crucial for proving the minimax optimality. When the relevance of source datasets is unknown, we develop an efficient detection procedure to identify informative sources and establish its selection consistency. Simulations and real data analysis are conducted to support the validity of our methodology.
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